Frequency tunable qubit control strategy

ABSTRACT

Methods, systems and apparatus for implementing a target two-qubit quantum logic gate on a first qubit and second qubit using a tunable qubit coupler. In one aspect, a method includes generating a control signal for the target two-qubit quantum logic gate according to a control model, wherein the control model comprises a controlled-Z operator and a swap operator that are non-orthogonal; and applying the control signal to the first qubit, second qubit and tunable qubit coupler.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority under 35 U.S.C. 119 to ProvisionalApplication No. 62/846,029, filed May 10, 2019, all of which areincorporated by reference.

BACKGROUND

This specification relates to quantum computing.

Classical computers have memories made up of bits, where each bit canrepresent either a zero or a one. Quantum computers maintain sequencesof quantum bits, called qubits, where each quantum bit can represent azero, one or any quantum superposition of zeros and ones. Quantumcomputers operate by setting qubits in an initial state and controllingthe qubits, e.g., according to a sequence of quantum logic gates.

SUMMARY

This specification describes control strategies for implementing quantumalgorithms using frequency tunable qubits.

In general, one innovative aspect of the subject matter described inthis specification can be implemented in a method for implementing atarget two-qubit quantum logic gate on a first qubit and second qubitusing a tunable qubit coupler, the method comprising: generating acontrol signal for the target two-qubit quantum logic gate according toa control model, wherein the control model comprises a controlled-Zoperator and a swap operator that are non-orthogonal; and applying thecontrol signal to the first qubit, second qubit and tunable qubitcoupler.

Other implementations of these aspect include corresponding computersystems, apparatus, and computer programs recorded on one or morecomputer storage devices, each configured to perform the actions of themethods. A system of one or more classical and/or quantum computers canbe configured to perform particular operations or actions by virtue ofhaving software, firmware, hardware, or a combination thereof installedon the system that in operation causes or cause the system to performthe actions. One or more computer programs can be configured to performparticular operations or actions by virtue of including instructionsthat, when executed by data processing apparatus, cause the apparatus toperform the actions.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations the control model further comprises one or more detuningoperators and a frequency shift operator.

In some implementations the control model is photon-conserving.

In same implementations the control model comprises a first detuningoperator multiplied by the swap operator multiplied by the controlled-Zoperator multiplied by a second detuning operator multiplied by thefrequency shift operator.

In some implementations the control model is given by

U=e ^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−a) ^(/4) e^(−i({circumflex over (X)}{circumflex over (X)}+ŶŶ)θ/2) e^(−i{circumflex over (Z)}{circumflex over (Z)}ϕ/4) e^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−b) ^(/4) e^(−i(Î{circumflex over (Z)}+{circumflex over (Z)}Î)δ) ⁺ ^(/4)

where Î represents the identity operator, {circumflex over (X)}represents the Pauli-X operator, Ŷ represents the Pauli-Y operator,{circumflex over (Z)} represents the Pauli-Z operator, δ_(−a), δ_(−b)represent detuning operation parameters, θ represents a swap angle and ϕrepresents a controlled-Z phase angle.

In some implementations generating the control signal for the targettwo-qubit quantum logic gate according to the control model comprises:determining values of the detuning operation parameters, swap angle andphase angle such that the control model represents the action of thetarget two-qubit quantum logic gate on the first qubit and the secondqubit; and generating the control signal based on the determined values.

In some implementations application of the control signal to the firstqubit, second qubit and tunable qubit coupler realizes the targettwo-qubit evolution with accuracy above a predetermined threshold.

In some implementations determining values of the detuning operationparameters, swap angle and phase angle comprises setting the detuningoperation parameters, swap angle and phase angle to respective valuesthat have been determined through a calibration procedure, thecalibration procedure comprising: for each set of values of qubitcontrol parameters from multiple sets of values of qubit controlparameters: performing a cross-entropy benchmarking experiment using theset of values for the qubit control parameters to identify a unitarytransformation realized by the set of values of qubit controlparameters, comprising determining values of the control modelparameters that maximize the fidelity of the identified unitarytransformation; storing data linking the set of values of qubit controlparameters, the unitary transformation the set of values realizes, andcorresponding determined values of control model parameters;interpolating over the stored data to determine unitary transformationsand the control model parameters corresponding to sets of values ofqubit control parameters that are not included in the multiple sets ofvalues of qubit control parameters.

In some implementations qubit control parameters comprise qubit voltage,qubit coupler voltage and pulse length.

In some implementations applying the control signal to the first qubit,the second qubit and the tunable qubit coupler comprises: applying afirst pulse to the first qubit and second qubit, wherein the first pulsedetunes the first qubit and second qubit; applying a second pulse to thetunable coupler that couples the first qubit and the second qubit,wherein the second pulse implements the swap operator; and applying athird pulse to the tunable coupler and one of the first qubit or secondqubit, wherein the third pulse implements the controlled-Z operator.

In some implementations applying a second pulse to the tunable couplerfurther comprises adjusting the frequency of the first and second qubitsuch that the first qubit and second qubit are on resonance.

In some implementations applying the third pulse to the tunable couplerand one of the first qubit or second qubit comprises adjusting thefrequency of the first qubit or second qubit.

In some implementations the first pulse comprises one or more squarepulses.

In some implementations the second pulse comprises a smooth pulse.

In some implementations the method further comprises implementing asequence of target two-qubit quantum logic gates, the method comprising:for each target two-qubit quantum logic gate: generating a controlsignal for the target two-qubit quantum logic gate according to acontrol model, wherein the control model comprises a controlled-Zoperator and a swap operator that are non-orthogonal; and applying thecontrol signal to a first qubit and a second qubit; and for a lasttarget two-qubit quantum logic gate in the sequence of target two-qubitquantum logic gates, measuring the first qubit and second qubit in the Zbasis.

In general, another innovative aspect of the subject matter described inthis specification can be implemented in a method for implementing atwo-qubit quantum logic gate using a first qubit, a second qubit and atunable qubit coupler, wherein implementing the two-qubit gate comprisesimplementing a controlled-Z operation and implementing a swap operation,wherein the controlled-Z operation and the swap operation arenon-orthogonal.

Other implementations of these aspect include corresponding computersystems, apparatus, and computer programs recorded on one or morecomputer storage devices, each configured to perform the actions of themethods. A system of one or more classical and/or quantum computers canbe configured to perform particular operations or actions by virtue ofhaving software, firmware, hardware, or a combination thereof installedon the system that in operation causes or cause the system to performthe actions. One or more computer programs can be configured to performparticular operations or actions by virtue of including instructionsthat, when executed by data processing apparatus, cause the apparatus toperform the actions.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations implementing the two-qubit quantum logic gate using afirst qubit, a second qubit and a qubit coupler comprises: applying afirst pulse to the first qubit and second qubit, wherein the first pulsedetunes the first qubit and second qubit; applying a second pulse to thetunable coupler that couples the first qubit and the second qubit,wherein the second pulse implements the swap operator; and applying athird pulse to the tunable coupler and one of the first qubit or secondqubit, wherein the third pulse implements the controlled-Z operator.

In some implementations applying the second pulse to the tunable couplerfurther comprises adjusting the frequency of the first and second qubitsuch that the first qubit and second qubit are on resonance.

In some implementations applying the third pulse to the tunable couplerand one of the first qubit or second qubit comprises adjusting thefrequency of the first qubit or second qubit.

In some implementations the first pulse comprises one or more squarepulses.

In some implementations the second pulse comprises a smooth pulse.

In some implementations the first pulse, second pulse, and third pulseare determined using a photon conserving control model given by

U=e ^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−a) ^(/4) e^(−i({circumflex over (X)}{circumflex over (X)}+ŶŶ)θ/2) e^(−i{circumflex over (Z)}{circumflex over (Z)}ϕ/4) e^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−b) ^(/4) e^(−i(Î{circumflex over (Z)}+{circumflex over (Z)}Î)δ) ⁺ ^(/4)

where Î represents the identity operator, {circumflex over (X)}represents the Pauli-X operator, Ŷ represents the Pauli-Y operator,{circumflex over (Z)} represents the Pauli-Z operator, δ_(−a), δ_(−b)represent detuning operation parameters, θ represents a swap angle forthe swap operator and ϕ represents a controlled-Z phase angle for thecontrolled-Z operator.

In some implementations determining the first pulse, second pulse andthird pulse using the control model comprises: determining values of thedetuning operation parameters, swap angle and phase angle such that thecontrol model represents the action of the target two-qubit quantumlogic gate on the first qubit and the second qubit.

In some implementations determining values of the detuning operationparameters, swap angle and phase angle comprises setting the detuningoperation parameters, swap angle and phase angle to respective valuesthat have been determined through a calibration procedure, thecalibration procedure comprising: for each set of values of qubitcontrol parameters from multiple sets of values of qubit controlparameters: performing a cross-entropy benchmarking experiment using theset of values for the qubit control parameters to identify a unitarytransformation realized by the set of values of qubit controlparameters, comprising determining values of the control modelparameters that maximize the fidelity of the identified unitarytransformation; storing data linking the set of values of qubit controlparameters, the unitary transformation the set of values realizes, andcorresponding determined values of control model parameters;interpolating over the stored data to determine unitary transformationsand the control model parameters corresponding to sets of values ofqubit control parameters that are not included in the multiple sets ofvalues of qubit control parameters.

In some implementations qubit control parameters comprise qubit voltage,qubit coupler voltage and pulse length.

In some implementations implementing the two-qubit quantum logic gatecomprises implementing the two-qubit quantum logic gate with accuracyabove a predetermined threshold.

The subject matter described in this specification can be implemented inparticular ways so as to realize one or more of the followingadvantages.

A system implementing individual or sequences of two-qubit quantum logicgates using the techniques described in this specification can providean improvement in the speed at which swap gates and controlled-z gatesare performed. For example, any swap gate and controlled-z gate(together with single qubit gates such as z rotations) can beimplemented in 35 ns. In contrast, other methods such as, e.g., thosethat decompose computations into controlled-Z and pi pulses, may requireseveral hundred nano seconds or more.

The control model and control sequences described in this specificationcan be used to efficiently generate any gate set required for near termapplications of quantum computers, e.g., variational eigensolvers,quantum approximate optimization algorithms or quantum supremacyexperiments, thus enabling the realization of such devices and improvingtheir functionality.

The details of one or more implementations of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts an example system for implementing two-qubit gates with atunable coupler.

FIG. 2 depicts an example implementation of a two-qubit quantum logicgate.

FIG. 3 shows numerical simulation results of swap gates.

FIG. 4 shows numerical simulation results of controlled-z gates.

FIG. 5 is a flow chart of an example process for implementing a targettwo-qubit quantum logic gate.

FIG. 6 is a flow chart of an example process for calibrating a swap gateor controlled-z gate.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION Overview

This specification describes experimental control strategies forimplementing quantum algorithms such as variational quantum eigensolvers(VQE)—a strategy for implementing quantum chemistry routines, quantumapproximate optimization algorithms (QAOA)—a strategy for implementingcombinatorial optimization algorithms on a quantum computer, or quantumsupremacy algorithms using frequency tunable qubits, e.g., transmonqubits.

Alternative control strategies implement quantum algorithms within thecontext of variational quantum eigensolvers and quantum approximateoptimization algorithms by decomposing the algorithms into single qubitgates, e.g., π and π/2 rotations, and two-qubit controlled-Z (CZ)quantum logic gates. However, these decompositions are extremelyinefficient. For example, implementing such decompositions withoutleakage to non-computational states is nearly impossible. In addition,implementing CZ gates typically introduces unwanted swap behavior. Toavoid such unwanted behavior and achieve high fidelity CZ gates, CZ gateexecution time is increased.

This specification describes methods and systems for efficient,low-leakage implementation of two-qubit quantum logic gates. As anexample, the described methods and systems enable efficientimplementation of the quantum gate sets required for algorithms such asVQE, QAOA and quantum supremacy.

A parameterized control model that describes any two-qubit evolutionthat conserves photon number, i.e., any two-qubit operation that aquantum computer may be required to perform, is presented. The controlmodel includes a swap operator and a CZ operator that are intentionallynon-orthogonal to each other. The control model can be used to generatea control signal that, when applied to two qubits, implements a targettwo-qubit quantum logic gate. The control signal includes a sequence ofcontrol pulses. A first control pulse detunes the first qubit and secondqubit. A second control pulse implements a swap quantum logic gate onthe first and second qubit. A third control pulse implements a CZquantum logic gate on the first and second qubit. By sacrificing localfidelity of the swap quantum logic gate and the CZ quantum logic gate,global fidelity and execution speed of the two-qubit evolution as awhole is increased. Quantum algorithms can therefore be more reliablyand efficiently executed.

Example Operating Environment

FIG. 1 is an example system that can perform the method described withreference to FIG. 5 .

The system 100 includes quantum hardware 102 that includes at least afirst qubit 104, a second qubit 106, and a tunable coupler 108 betweenthe first qubit and the second qubit. The first qubit 104, the secondqubit 106, and the tunable coupler 108 may be subcomponents of thequantum hardware 102. For example, quantum hardware 102 may includeadditional components for performing quantum or classical computations,e.g., additional qubits, additional tunable couplers, additional controlelectronics and processors.

Each of the first qubit 104, the second qubit 106, and the tunablecoupler 108 are frequency-tunable. In some implementations the firstqubit 104 and the second qubit 106 may be superconducting qubits. Forexample, the first qubit 104 and the second qubit 106 may be transmonqubits. In other implementations other qubit architectures may be usedinstead.

Various different tunable coupler designs may be used. For example, insome implementations the tunable coupler 108 may include athree-terminal device constructed from superconductor materials using afixed negative mutual inductance and a single, current-biased Josephsonjunction that acts as a tunable positive inductance. Further discussionand examples of tunable couplers are described in detail in “A tunablecoupling scheme for implementing high-fidelity two-qubit gates,” Fei Yanet al., arxiv:quant-ph/180309813v1, “Demonstration of a Tuneable Couplerfor Superconducting Qubits Using Coherent, Time Domain, Two-QubitOperations,” R. C. Bialczak et al., arxiv: quant-ph/1007.2219v1, “Sign-and magnitude-tunable coupler for superconducting flux qubits,” and R.Harris et al., arxiv:cond-mat/0608253v4, “Tunable coupler forsuperconducting Xmon qubits: Perturbative nonlinear model,” Michael R.Geller et al., arxiv:quant-ph/1405.1915v1, each of which is incorporatedherein by reference in its entirety.

The system 100 includes control electronics 110. Control electronics 110includes control devices that may operate the quantum hardware. Forexample, control electronics 110 may include an arbitrary waveformgenerator.

The system 100 includes qubit control lines 112 from the controlelectronics 110 to the first qubit 104 and the second qubit 106,respectively. For example, the frequency of the first qubit 104 and thesecond qubit 106 can be tuned using qubit control lines 112. Thefrequency of the first qubit 104 and the second qubit 106 may be tunedby applying control signals to the qubit control lines 112 via controlelectronics 110. In addition, control electronics 110 can performmeasurements of the first qubit 104 and the second qubit 106 throughqubit control lines 112. Measurement of the first qubit 104 andmeasurement of the second qubit 106 determines the state of the firstqubit 104 and the second qubit 106, respectively. Control electronics110 can store, display, and/or further process the results of each ofthe measurements of the first qubit 104 and the second qubit 106.

The system 100 includes tunable coupler control line 114. Controlelectronics 110 can dynamically tune the coupling or interaction betweenthe first qubit 104 and the second qubit 106 by applying control signalsto the tunable coupler control line 114 to tune the tunable coupler 108frequency. For example, control electronics 110 may apply a voltagepulse to the tunable coupler control line 114 to tune the tunablecoupler 108 frequency. In some implementations, the control electronics110 may include a data processing apparatus and associated memory. Thememory may include a computer program having instructions that, whenexecuted by the data processing apparatus, cause the data processingapparatus to perform one or more functions described herein, such asapplying a control signal to a qubit and/or to a tunable coupler.

FIG. 2 is an illustration 200 of an example implementation of atwo-qubit quantum logic gate. The illustration 200 is an example of animplementation of a two-qubit quantum logic gate performed by the system100 of FIG. 1 .

In the illustration 200, a two-qubit quantum logic gate is applied to afirst qubit 202 and a second qubit 204 through application of a controlsignal to the first qubit 202, second qubit 204 and a tunable coupler206 coupling the first qubit 202 and second qubit 204. For example, thefirst qubit 202, second qubit 204, tunable coupler 206 and controlsignal may correspond to the first qubit 104, second qubit 106, tunablecoupler 108 and control signal generated by the control electronics 110and applied via the tunable coupler control line 114 of FIG. 1 .Throughout this specification, application of a control signal isunderstood to include application of a collection of multiple controlsignals, in which individual control signals are applied to either thefirst qubit, the second qubit or the coupler.

The control signal is a generated according to a control model. Thecontrol model represents a unitary transformation that describes any twoqubit evolution with low leakage. In some implementations the controlmodel may be photon conserving—an important property in quantumchemistry where particle number must be conserved during evolution. Thecontrol model includes a product of terms, including a controlled-Zoperator and a swap operator.

A controlled-Z operator is a quantum logic gate that acts on two or morequbits, where one or more qubits act as a control for a Pauli-Zoperation. For example, a two-qubit controlled-Z operator is a quantumlogic gate that acts on two qubits, where a first qubit acts as acontrol, and performs a Pauli-Z operation on the second qubit only whenthe first qubit is in the 1 state. With respect to the basis 00, 01, 10,11 a two-qubit controlled-Z operator may be represented by the matrix

${C(Z)} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & {- 1}\end{pmatrix}$

The action of the two-qubit controlled-Z operator is to add a phase tothe 11 state when the first qubit is in the 1 state.

A swap operator is a quantum logic gate that acts on two qubits andswaps the states of two qubits. With respect to the basis 00, 01, 10, 11a swap operator may be represented by the matrix

${SWAP} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\end{pmatrix}$

The action of the swap operator is to exchange the states 01↔10.

Implementing high fidelity controlled-Z and swap operations, e.g., usinghardware similar or identical to that described with reference to FIG. 1, can be difficult. For example, if a controlled-Z operation isperformed too quickly, unwanted swap behavior can occur. That is,instead of only adding a phase to the 11 state, performing thecontrolled-Z operation too quickly may result in addition of a phase tothe 11 state and exchange of the states 10 and 01. Similarly, if a swapoperation is performed too quickly, unwanted phases may accumulate onthe 11 state. That is, instead of performing a clean exchange of thestates 01 and 10, the action of the swap operation may be an exchange ofthe states 10 and 01 and an addition of a phase on the 11 state. Toovercome these difficulties, controlled-Z operations and swap operationsare typically performed slowly, e.g., taking several hundrednanoseconds. This in turn results in slow algorithm execution times.

The control model described in this specification includes acontrolled-Z operator and a swap operator. The controlled-Z operator andswap operators are non-orthogonal. Therefore, application of thecontrolled-Z operator intentionally incurs swap behavior, andapplication of the swap operator intentionally adds a phase to the 11state. Implementing the controlled-Z operators and swap operatorstherefore takes less time than high-fidelity controlled-Z and swapoperators, e.g., 20 or 14 nanoseconds, as described in more detail belowwith reference to FIGS. 3 and 4 .

The control model may also include one or more detuning operators and afrequency shift operator. For example, the control model may be given by

$U = {\underset{Detuning}{\underset{︸}{e^{- {i({{\hat{I}\hat{Z}} - {\hat{Z}\hat{I}}})}\delta_{- a}/4}}}\underset{{Approximate}{swap}}{\underset{︸}{e^{- {i({{\hat{X}\hat{X}} + {\hat{Y}\hat{Y}}})}\theta/2}}}\underset{{Approximate}{CZ}}{\underset{︸}{e^{- i\hat{Z}\hat{Z}\phi/4}}}\underset{Detuning}{\underset{︸}{e^{- {i({{\hat{I}\hat{Z}} - {\hat{Z}\hat{I}}})}\delta_{- b}/4}}}\underset{{Frequency}{shift}}{\underset{︸}{e^{- {i({{\hat{I}\hat{Z}} + {\hat{Z}\hat{I}}})}\delta_{+}/4}}}}$

where Î represents the identity operator, {circumflex over (X)}represents the Pauli-X operator, Ŷ represents the Pauli-Y operator,{circumflex over (Z)} represents the Pauli-Z operator, δ_(−a), δ_(−b)represent detuning operation parameters, θ represents a swap angle, ϕrepresents a controlled-Z phase angle and δ_(−b) represents a frequencyshift parameter. For notational conciseness, in the above equation thefirst operator in a product of Pauli operators is taken to act on thefirst qubit and the second operator in a product of two operators istaken to act on the second qubit. For example, the product IZ meansapply an identity operator to the first qubit and a Pauli Z operator tothe second qubit. Equivalently, the products may be re-written as, e.g.X₁X₂ where the suffix indicates which of the qubits the operator isacting on.

The control model U describes any operation that the quantum hardwareand control electronics of FIG. 1 can perform. For example, the controlmodel U may describe gate sets required for VQE, QAOA and quantumsupremacy experiments. Therefore, to implement a target two qubitquantum logic gate with an accuracy above a predetermined threshold,values of the detuning operation parameters, swap angle, phase angle andfrequency shift parameter that generate an instance of the control modelU representing the action of the target two-qubit quantum logic gate ontwo qubits are determined, e.g., through the calibration processdescribed below with reference to FIG. 6 . The values are then used togenerate a control signal that is applied to the two qubits 202, 204 andthe tunable coupler 206.

The control signal includes a sequence of pulses, e.g., pulses 208 a-ethat are applied to the first qubit 202, second qubit 204 and tunablecoupler 206. The sequence of pulses includes a first pulse 208 a andsecond pulse 208 b that detune the first qubit 202 and the second qubit204. That is, the first pulse 208 a sets the operating frequency of thefirst qubit 202 to a first value. The second pulse 208 b sets theoperating frequency of the second qubit 204 to a second value, where thefirst value and second value are separated by a predetermined distancethat does not allow the qubits 202, 204 to interact with each other(i.e. there is no, or a negligible, interaction between the qubits). Thespecific values are dependent on the physical realizations of the qubits202, 204 and may vary. As illustrated in FIG. 2 , the shape of the firstpulse 208 a and the second pulse 208 b may be square to allow for fastqubit detuning. In some cases the pulses 208 a and 208 b can be appliedapproximately simultaneously, e.g., as a single control pulse.

The sequence of pulses includes a third pulse 208 c that is applied tothe tunable coupler 206 and that implements the swap operator. When thethird pulse 208 c is applied to the tunable coupler 206, the first qubit202 and second qubit 204 are on resonance. Generally, in FIG. 2 thequbits can be assumed to be on resonance and only move out of resonancewhen a voltage is applied to the qubits. At step 210 and 212 no pulse isapplied and the qubits are therefore assumed to be resonant during theswap-like gate. As illustrated in FIG. 2 , the shape of the second pulse208 b may be smoothed, e.g., by a Gaussian filter with a bandwidthranging from 60 MHz to 100 MHz, to avoid leakage.

The sequence of pulses includes a fourth pulse 208 d that is applied tothe first qubit 202 to set the operating frequency of the first qubit202 to a third value. For convenience, the first, second and thirdvalues represented by (the amplitudes of the) pulses 208 a, 208 b and208 d are illustrated in FIG. 2 as being different values, however thevalues may vary and in some cases may be the same. As illustrated inFIG. 2 , the shape of the fourth pulse may be square to allow for fastsetting of the first qubit operating frequency.

The sequence of pulses includes a fifth pulse 208 e that is applied tothe tunable coupler 206 and that implements the controlled-Z operator.When the fifth pulse 208 e is applied to the tunable coupler 206, thedifference between the first qubit 202 and second qubit 204 frequenciesis a fixed value. This difference can be achieved by detuning one of thequbits, the other qubit, or both qubits. As illustrated in FIG. 2 , theshape of the fifth pulse 208 d may be smooth to avoid leakage. In somecases the pulses 208 d and 208 e can be applied approximatelysimultaneously, e.g., as a single control pulse.

The sequence of pulses 208 a-e can be separated into three stages. Thefirst stage is a detuning stage 216 that includes application of pulses208 a and 208 b and corresponds to the detuning operators included inthe control model U. The second stage is a swap stage 218 that includesapplication of pulse 208 c and corresponds to the swap operator includedin the control model U. The third stage 220 is a CZ stage 220 thatincludes application of pulses 208 d and 208 e and corresponds to the CZoperator included in the control model U. In each stage, pulses in thestage can be applied approximately simultaneously.

The sequence of pulses 208 a-e can correspond to a first cycle 208 inone of multiple cycles 208, 222, where the first cycle 208 representsimplementation of a first quantum logic gate in a sequence of multiplequantum logic gates that performs a specific computation or algorithm.When multiple cycles of sequences of pulses are implemented, the lastfrequency shift operator in the control model for a preceding two-qubitquantum logic gate in the sequence can be combined with the firstdetuning operator in the control model for a current two-qubit quantumlogic gate in the sequence. This is why the sequence of pulses 208 a-edoes not include a frequency shift stage.

When multiple cycles of a sequence of pulses are performed, the qubits202, 204 may be measured in the Z-basis. This negates the requirement ofimplementing the final frequency shift operator in the control modelcorresponding to a final quantum logic gate in the sequence of quantumlogic gates. Alternatively or in addition, one or more single qubitrotation operations may be implemented on the first qubit 202 and secondqubit 204 prior to measurement. In other words, the control model Uincludes 5 terms but the pulse sequence shown in FIG. 2 has 3 sections216, 218, and 220. When constructing a long sequence of gates,neighboring z-rotations can be combined. This produces 3 termseverywhere except for at the end where an additional z-rotation remains,but this can be ignored when measuring in the z-basis.

FIG. 3 shows numerical simulation results of how to apply swap gates.Two plots are shown in FIG. 3 . The first plot 300 includes an x axisrepresenting the length of a control pulse (measured in ns) applied toimplement a swap operation, e.g., the length of pulse 208 c of FIG. 2 .They axis represents a maximum coupling strength (measured in MHz on acorresponding coupling wire) of the two qubits that are to undergo theswap operation. The plot 300 shows the different swap angles ϕ (asdefined in control model U described above with reference to FIG. 2 )that can be achieved for different values of pulse length T and maximumqubit coupling g, where ϕ, T, g are related via

$\phi = {\frac{3}{4}\pi{T\left( {\eta + \sqrt{{8{\mathcal{g}}^{2}} + \eta^{2}}} \right)}}$

with η representing qubit non-linearity. In plots 300 and 350, η isassumed to equal −200 MHz, it is assumed that the qubits are onresonance and that the control pulse is smooth, e.g., a raised cosinepulse. However, these assumptions are based on example hardwareconsiderations, and may vary.

The plot 300 shows that by choosing control pulse duration equal to 14ns (see vertical line 302), all swap angles between 0.0 and 1.5 can beachieved by sweeping the maximum voltage on the coupler wire from avoltage corresponding to a coupling of 0 MHz to a voltage correspondingto a coupling of 50 MHz. It is noted that other pulse durations can alsoachieve all swap angles between 0.0 and 1.5, however 14 ns is the lowerbound in this example. White areas of the plot 300 represent areas ofhigh leakage (of population or energy to non-computational basisstates), e.g., leakage>0.1%.

The second plot 350 includes an x axis representing the length of acontrol pulse (measured in ns) applied to implement a swap operation,e.g., the length of pulse 208 c of FIG. 2 . They axis represents amaximum coupling strength (measured in MHz) of the two qubits that areto undergo the swap operation. The plot 350 shows accumulation ofunwanted CZ behavior—conditional phases θ (as defined in control model Udescribed above with reference to FIG. 2 )—that can be incurred fordifferent values of pulse length and maximum qubit coupling duringimplementation of the swap operation of plot 300.

The second plot 350 shows that the swap operation of plot 300 is a swapoperation—it is non-orthogonal to a controlled-z operation because theconditional phase θ is incurred at all pulse lengths when sweeping themaximum voltage on the coupler wire from a voltage corresponding to acoupling of 0 MHz to a voltage corresponding to a coupling of 50 MHz.However, at a fixed pulse length of 14 ns the conditional phaseaccumulation is minimal.

Both plots 300 and 350 show how, under the above described assumptions,any swap angle can be implemented in 14 ns with a maximum couplingstrength between 0 and 50 MHz with minimal leakage.

FIG. 4 shows numerical simulation results of how to apply controlled-Zgates. Two plots are shown in FIG. 4 . The first plot 400 includes an xaxis representing the detuning of the two qubits when a controlled-zoperation is implemented (measured in MHz), e.g., the detuning of qubit202 and 204 described with reference to FIG. 2 . The y axis represents amaximum coupling strength (measured in MHz on a corresponding couplingwire) of the two qubits that are to undergo the controlled-z operation.In plots 400 and 450, η is assumed to equal −200 MHz, the pulse lengthused to implement the controlled-z operation is assumed to equal 20 nsand be smooth, e.g., a raised cosine pulse. However, these assumptionsare based on example hardware considerations, and may vary.

Plot 400 shows unwanted swap angles ϕ that can be incurred for differentvalues of pulse length and maximum qubit coupling during implementationof a controlled-z operation. As shown in plot 400, low detuning valuesof 60 MHz or less incur the most unwanted swap behavior.

Plot 450 shows that all conditional phase angles θ can be achievedbetween detuning values of 120 MHz and 300 MHz. Therefore, by choosingvalues of detuning and maximum coupling according to the parabolic line402, any conditional phase angle can be implemented in 20 ns withminimal swap behavior.

Combining the results of FIGS. 3 and 4 shows how using the techniquesdescribed in this specification, namely the control model U and controlsequence shown in FIG. 2 , a combination of any swap angle and anyconditional phase angle can be implemented in less than 35 ns.

Example Method for Implementing a Target Two-Qubit Quantum Logic Gate

FIG. 5 is a flow diagram of an example process 500 for implementing atarget two-qubit quantum logic gate. For convenience, the process 500will be described as being performed by quantum hardware incommunication with control electronics located in one or more locations.For example, the system 100 of FIG. 1 , appropriately programmed inaccordance with this specification, can perform the process 500.

A control signal for the target two-qubit quantum logic gate isgenerated according to a control model (step 502). The control modelincludes a controlled-Z operator and a swap operator that arenon-orthogonal. The control model may further include one or moredetuning operators and a frequency shift operator. For example, thecontrol model may include a first detuning operator multiplied by theswap operator multiplied by the controlled-Z operator multiplied by asecond detuning operator multiplied by the frequency shift operator. Anexample expression for the control model is given above with referenceto FIG. 2 , and for convenience is repeated here:

U=e ^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−a) ^(/4) e^(−i({circumflex over (X)}{circumflex over (X)}+ŶŶ)θ/2) e^(−i{circumflex over (Z)}{circumflex over (Z)}ϕ/4) e^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−b) ^(/4) e^(−i(Î{circumflex over (Z)}+{circumflex over (Z)}Î)δ) ⁺ ^(/4)

where Î represents the identity operator, {circumflex over (X)}represents the Pauli-X operator, Ŷ represents the Pauli-Y operator,{circumflex over (Z)} represents the Pauli-Z operator, δ_(−a), δ_(−b)represent detuning operation parameters, θ represents a swap angle and ϕrepresents a controlled-Z phase angle. In some cases the control modelis photon conserving.

Generating the control signal may include determining values of thedetuning operation parameters, swap angle and phase angle to respectivevalues such that the control model represents the action of the targettwo-qubit quantum logic gate on a first qubit and a second qubit, andgenerating the control signal based on the determined values. Specificvalues of the detuning operation parameters, swap angle and phase anglecan be determined using the calibration process described below withreference to FIG. 6 .

The generated control signal is applied to the first qubit and secondqubit to realize the target two-qubit evolution with accuracy above apredetermined threshold (step 504). Applying the control signal mayinclude applying pulses to the first qubit and second qubit to detunethe first qubit and second qubit. The first pulse may include one ormore square pulses. A second pulse may be applied to a tunable couplerthat couples the first qubit and the second qubit, where the secondpulse implements the swap operator. When the second pulse is applied,the frequency of the first qubit and second qubit may adjusted such thatthe first qubit and second qubit are on resonance. In someimplementations the second pulse may include a smooth pulse. A thirdpulse may be applied to the tunable coupler and one of the first qubitor second qubit, where the third pulse implements the controlled-Zoperator. When the third pulse is applied to the tunable coupler and oneof the first qubit or second qubit, the frequency of the first qubit orsecond qubit may also be adjusted.

The process 500 may be repeated to implement a sequence of targettwo-qubit quantum logic gates. In these implementations the steps 502and 504 may be repeated for each two-qubit quantum logic gate in thesequence. At the end of the sequence, i.e., for a last target two-qubitquantum logic gate in the sequence of target two-qubit quantum logicgates, the first qubit and second qubit may be measured in the Z basisto obtain a result of the computation corresponding to the sequence oftwo-qubit quantum logic gates.

FIG. 6 is a flow diagram of an example process 600 for calibratingcontrol model parameters. For convenience, the process 600 will bedescribed as being performed by quantum hardware in communication withcontrol electronics located in one or more locations. For example, thesystem 100 of FIG. 1 , appropriately programmed in accordance with thisspecification, can perform the process 600.

For each set of values of qubit control parameters, e.g., qubit voltage,coupler voltage and pulse length, from multiple sets of values of qubitcontrol parameters, a cross-entropy benchmarking experiment is performedto identify a unitary transformation realized by the set of values ofqubit control parameters (step 602).

As part of the cross entropy benchmarking experiment, values of controlmodel parameters, e.g., values of the detuning operation parameters,swap angle and phase angle, that maximize the fidelity of the identifiedunitary transformation realized by the set of values of qubit controlparameters are determined. For example, the length of the pulses can bedetermined based on data similar to that shown in FIG. 3 . Referring toFIG. 3 , it can be seen that for pulses>14 ns, it is possible to achieveall swap angles while avoided leakage (the white regions representerrors). The amplitudes of the pulses can be determined based on thedata similar to that shown in FIG. 3 and FIG. 4 . Referring to FIGS. 3and 4 , the axes labeled ‘max coupling’ could be varied by changing thepulse height of the coupler pulses, the axes labeled ‘detuning’ could bevaried by changing the pulse height of the qubit pulses. These plotscould then be used to determine a mapping between pulse heights and gateparameters. That is, scanning values of pulse lengths and pulse heights,and measuring the model parameters (using cross-entropy benchmarking),provides a map between pulse parameters and control model parameters.

Data linking the set of values of qubit control parameters, the unitarytransformation the set of values realizes, and corresponding determinedvalues of control model parameters is stored (604).

Unitary transformations and control model parameters corresponding tosets of values of qubit control parameters that are not included in themultiple sets of values of qubit control parameters are determined byinterpolating over the stored data.

Implementations of the subject matter and operations described in thisspecification can be implemented in digital electronic circuitry, analogelectronic circuitry, suitable quantum circuitry or, more generally,quantum computational systems, in tangibly-embodied software orfirmware, in computer hardware, including the structures disclosed inthis specification and their structural equivalents, or in combinationsof one or more of them. The term “quantum computational systems” mayinclude, but is not limited to, quantum computers, quantum informationprocessing systems, quantum cryptography systems, or quantum simulators.

Implementations of the subject matter described in this specificationcan be implemented as one or more computer programs, i.e., one or moremodules of computer program instructions encoded on a tangiblenon-transitory storage medium for execution by, or to control theoperation of, data processing apparatus. The computer storage medium canbe a machine-readable storage device, a machine-readable storagesubstrate, a random or serial access memory device, one or more qubits,or a combination of one or more of them. Alternatively or in addition,the program instructions can be encoded on an artificially-generatedpropagated signal that is capable of encoding digital and/or quantuminformation, e.g., a machine-generated electrical, optical, orelectromagnetic signal, that is generated to encode digital and/orquantum information for transmission to suitable receiver apparatus forexecution by a data processing apparatus.

The terms quantum information and quantum data refer to information ordata that is carried by, held or stored in quantum systems, where thesmallest non-trivial system is a qubit, i.e., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that may be suitably approximated as atwo-level system in the corresponding context. Such quantum systems mayinclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, inns orsuperconducting qubits. In many implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states are possible.

The term “data processing apparatus” refers to digital and/or quantumdata processing hardware and encompasses all kinds of apparatus,devices, and machines for processing digital and/or quantum data,including by way of example a programmable digital processor, aprogrammable quantum processor, a digital computer, a quantum computer,multiple digital and quantum processors or computers, and combinationsthereof. The apparatus can also be, or further include, special purposelogic circuitry, e.g., an FPGA (field programmable gate array), an ASIC(application-specific integrated circuit), or a quantum simulator, i.e.,a quantum data processing apparatus that is designed to simulate orproduce information about a specific quantum system. In particular, aquantum simulator is a special purpose quantum computer that does nothave the capability to perform universal quantum computation. Theapparatus can optionally include, in addition to hardware, code thatcreates an execution environment for digital and/or quantum computerprograms, e.g., code that constitutes processor firmware, a protocolstack, a database management system, an operating system, or acombination of one or more of them.

A digital computer program, which may also be referred to or describedas a program, software, a software application, a module, a softwaremodule, a script, or code, can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A quantum computerprogram, which may also be referred to or described as a program,software, a software application, a module, a software module, a script,or code, can be written in any form of programming language, includingcompiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language,or can be written in a quantum programming language, e.g., QCL orQuipper.

A computer program may, but need not, correspond to a file in a filesystem. A program can be stored in a portion of a file that holds otherprograms or data, e.g., one or more scripts stored in a markup languagedocument, in a single file dedicated to the program in question, or inmultiple coordinated files, e.g., files that store one or more modules,sub-programs, or portions of code. A computer program can be deployed tobe executed on one computer or on multiple computers that are located atone site or distributed across multiple sites and interconnected by adigital and/or quantum data communication network. A quantum datacommunication network is understood to be a network that may transmitquantum data using quantum systems, e.g. qubits. Generally, a digitaldata communication network cannot transmit quantum data, however aquantum data communication network may transmit both quantum data anddigital data.

The processes and logic flows described in this specification can beperformed by one or more programmable computers, operating with one ormore processors, as appropriate, executing one or more computer programsto perform functions by operating on input data and generating output.The processes and logic flows can also be performed by, and apparatuscan also be implemented as, special purpose logic circuitry, e.g., anFPGA or an ASIC, or a quantum simulator, or by a combination of specialpurpose logic circuitry or quantum simulators and one or more programmeddigital and/or quantum computers.

For a system of one or more computers to be “configured to” performparticular operations or actions means that the system has installed onit software, firmware, hardware, or a combination of them that inoperation cause the system to perform the operations or actions. For oneor more computer programs to be configured to perform particularoperations or actions means that the one or more programs includeinstructions that, when executed by data processing apparatus, cause theapparatus to perform the operations or actions. For example, a quantumcomputer may receive instructions from a digital computer that, whenexecuted by the quantum computing apparatus, cause the apparatus toperform the operations or actions.

Computers suitable for the execution of a computer program can be basedon general or special purpose processors, or any other kind of centralprocessing unit. Generally, a central processing unit will receiveinstructions and data from a read-only memory, a random access memory,or quantum systems suitable for transmitting quantum data, e.g. photons,or combinations thereof.

The elements of a computer include a central processing unit forperforming or executing instructions and one or more memory devices forstoring instructions and digital, analog, and/or quantum data. Thecentral processing unit and the memory can be supplemented by, orincorporated in, special purpose logic circuitry or quantum simulators.Generally, a computer will also include, or be operatively coupled toreceive data from or transfer data to, or both, one or more mass storagedevices for storing data, e.g., magnetic, magneto-optical disks, opticaldisks, or quantum systems suitable for storing quantum information.However, a computer need not have such devices.

Quantum circuit elements (also referred to as quantum computing circuitelements) include circuit elements for performing quantum processingoperations. That is, the quantum circuit elements are configured to makeuse of quantum-mechanical phenomena, such as superposition andentanglement, to perform operations on data in a non-deterministicmanner. Certain quantum circuit elements, such as qubits, can beconfigured to represent and operate on information in more than onestate simultaneously. Examples of superconducting quantum circuitelements include circuit elements such as quantum LC oscillators, qubits(e.g., flux qubits, phase qubits, or charge qubits), and superconductingquantum interference devices (SQUIDs) (e.g., RF-SQUID or DC-SQUID),among others.

In contrast, classical circuit elements generally process data in adeterministic manner. Classical circuit elements can be configured tocollectively carry out instructions of a computer program by performingbasic arithmetical, logical, and/or input/output operations on data, inwhich the data is represented in analog or digital form. In someimplementations, classical circuit elements can be used to transmit datato and/or receive data from the quantum circuit elements throughelectrical or electromagnetic connections. Examples of classical circuitelements include circuit elements based on CMOS circuitry, rapid singleflux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices andERSFQ devices, which are an energy-efficient version of RSFQ that doesnot use bias resistors.

In certain cases, some or all of the quantum and/or classical circuitelements may be implemented using, e.g., superconducting quantum and/orclassical circuit elements. Fabrication of the superconducting circuitelements can entail the deposition of one or more materials, such assuperconductors, dielectrics and/or metals. Depending on the selectedmaterial, these materials can be deposited using deposition processessuch as chemical vapor deposition, physical vapor deposition (e.g.,evaporation or sputtering), or epitaxial techniques, among otherdeposition processes. Processes for fabricating circuit elementsdescribed herein can entail the removal of one or more materials from adevice during fabrication. Depending on the material to be removed, theremoval process can include, e.g., wet etching techniques, dry etchingtechniques, or lift-off processes. The materials forming the circuitelements described herein can be patterned using known lithographictechniques (e.g., photolithography or e-beam lithography).

During operation of a quantum computational system that usessuperconducting quantum circuit elements and/or superconductingclassical circuit elements, such as the circuit elements describedherein, the superconducting circuit elements are cooled down within acryostat to temperatures that allow a superconductor material to exhibitsuperconducting properties. A superconductor (alternativelysuperconducting) material can be understood as material that exhibitssuperconducting properties at or below a superconducting criticaltemperature. Examples of superconducting material include aluminum(superconductive critical temperature of 1.2 kelvin) and niobium(superconducting critical temperature of 9.3 kelvin). Accordingly,superconducting structures, such as superconducting traces andsuperconducting ground planes, are formed from material that exhibitssuperconducting properties at or below a superconducting criticaltemperature.

In certain implementations, control signals for the quantum circuitelements (e.g., qubits and qubit couplers) may be provided usingclassical circuit elements that are electrically and/orelectromagnetically coupled to the quantum circuit elements. The controlsignals may be provided in digital and/or analog form.

Computer-readable media suitable for storing computer programinstructions and data include all forms of non-volatile digital and/orquantum memory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems,e.g., trapped atoms or electrons. It is understood that quantum memoriesare devices that can store quantum data for a long time with highfidelity and efficiency, e.g., light-matter interfaces where light isused for transmission and matter for storing and preserving the quantumfeatures of quantum data such as superposition or quantum coherence.

Control of the various systems described in this specification, orportions of them, can be implemented in a computer program product thatincludes instructions that are stored on one or more non-transitorymachine-readable storage media, and that are executable on one or moreprocessing devices. The systems described in this specification, orportions of them, can each be implemented as an apparatus, method, orsystem that may include one or more processing devices and memory tostore executable instructions to perform the operations described inthis specification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub-combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular implementations of the subject matter have been described.Other implementations are within the scope of the following claims. Forexample, the actions recited in the claims can be performed in adifferent order and still achieve desirable results. As one example, theprocesses depicted in the accompanying figures do not necessarilyrequire the particular order shown, or sequential order, to achievedesirable results. In some cases, multitasking and parallel processingmay be advantageous.

What is claimed is:
 1. A method for implementing a target two-qubitquantum logic gate on a first qubit and second qubit using a tunablequbit coupler, the method comprising: generating a control signal forthe target two-qubit quantum logic gate according to a control model,wherein the control model comprises a controlled-Z operator and a swapoperator that are non-orthogonal; and applying the control signal to thefirst qubit, second qubit and tunable qubit coupler.
 2. The method ofclaim 1, wherein the control model further comprises one or moredetuning operators and a frequency shift operator.
 3. The method ofclaim 1, wherein the control model is photon-conserving.
 4. The methodof claim 2, wherein the control model comprises a first detuningoperator multiplied by the swap operator multiplied by the controlled-Zoperator multiplied by a second detuning operator multiplied by thefrequency shift operator.
 5. The method of claim 1, wherein the controlmodel is given byU=e ^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−a) ^(/4) e^(−i({circumflex over (X)}{circumflex over (X)}+ŶŶ)θ/2) e^(−i{circumflex over (Z)}{circumflex over (Z)}ϕ/4) e^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−b) ^(/4) e^(−i(Î{circumflex over (Z)}+{circumflex over (Z)}Î)δ) ⁺ ^(/4) where Îrepresents the identity operator, {circumflex over (X)} represents thePauli-X operator, Ŷ represents the Pauli-Y operator, {circumflex over(Z)} represents the Pauli-Z operator, δ_(−a), δ_(−b) represent detuningoperation parameters, θ represents a swap angle and ϕ represents acontrolled-Z phase angle.
 6. The method of claim 4, wherein generatingthe control signal for the target two-qubit quantum logic gate accordingto the control model comprises: determining values of detuning operationparameters, a swap angle and a phase angle such that the control modelrepresents the action of the target two-qubit quantum logic gate on thefirst qubit and the second qubit; and generating the control signalbased on the determined values.
 7. The method of claim 1, whereinapplication of the control signal to the first qubit, second qubit andtunable qubit coupler realizes a target two-qubit evolution thatrepresents the target two-qubit quantum logic gate with accuracy above apredetermined threshold.
 8. The method of claim 6, wherein determiningvalues of the detuning operation parameters, swap angle and phase anglecomprises setting the detuning operation parameters, swap angle andphase angle to respective values that have been determined through acalibration procedure, the calibration procedure comprising: for eachset of values of qubit control parameters from multiple sets of valuesof qubit control parameters: performing a cross-entropy benchmarkingexperiment using the set of values for the qubit control parameters toidentify a unitary transformation realized by the set of values of qubitcontrol parameters, comprising determining values of the control modelparameters that maximize the fidelity of the identified unitarytransformation; storing data linking the set of values of qubit controlparameters, the unitary transformation the set of values realizes, andcorresponding determined values of control model parameters;interpolating over the stored data to determine unitary transformationsand the control model parameters corresponding to sets of values ofqubit control parameters that are not included in the multiple sets ofvalues of qubit control parameters.
 9. The method of claim 8, whereinqubit control parameters comprise qubit voltage, qubit coupler voltageand pulse length.
 10. The method of claim 1, wherein applying thecontrol signal to the first qubit, the second qubit and the tunablequbit coupler comprises: applying a first pulse to the first qubit andsecond qubit, wherein the first pulse detunes the first qubit and secondqubit; applying a second pulse to the tunable coupler that couples thefirst qubit and the second qubit, wherein the second pulse implementsthe swap operator; and applying a third pulse to the tunable coupler andone of the first qubit or second qubit, wherein the third pulseimplements the controlled-Z operator.
 11. The method of claim 10,wherein applying a second pulse to the tunable coupler further comprisesadjusting the frequency of the first and second qubit such that thefirst qubit and second qubit are on resonance.
 12. The method of claim10, wherein applying the third pulse to the tunable coupler and one ofthe first qubit or second qubit comprises adjusting the frequency of thefirst qubit or second qubit.
 13. The method of claim, 10, wherein thefirst pulse comprises one or more square pulses.
 14. The method of claim10, wherein the second pulse comprises a smooth pulse.
 15. (canceled)16. A method comprising: implementing a two-qubit quantum logic gateusing a first qubit, a second qubit and a tunable qubit coupler, whereinimplementing the two-qubit gate comprises implementing a controlled-Zoperation and implementing a swap operation, wherein the controlled-Zoperation and the swap operation are non-orthogonal.
 17. The method ofclaim 16, wherein implementing the two-qubit quantum logic gate using afirst qubit, a second qubit and a qubit coupler comprises: applying afirst pulse to the first qubit and second qubit, wherein the first pulsedetunes the first qubit and second qubit; applying a second pulse to thetunable coupler that couples the first qubit and the second qubit,wherein the second pulse implements the swap operator; and applying athird pulse to the tunable coupler and one of the first qubit or secondqubit, wherein the third pulse implements the controlled-Z operator. 18.The method of claim 17, wherein applying the second pulse to the tunablecoupler further comprises adjusting the frequency of the first andsecond qubit such that the first qubit and second qubit are onresonance.
 19. The method of claim 17, wherein applying the third pulseto the tunable coupler and one of the first qubit or second qubitcomprises adjusting the frequency of the first qubit or second qubit.20. The method of claim, 17, wherein the first pulse comprises one ormore square pulses.
 21. The method of claim 17, wherein the second pulsecomprises a smooth pulse.
 22. The method of claim 17, wherein the firstpulse, second pulse, and third pulse are determined using a photonconserving control model given byU=e ^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−a) ^(/4) e^(−i({circumflex over (X)}{circumflex over (X)}+ŶŶ)θ/2) e^(−i{circumflex over (Z)}{circumflex over (Z)}ϕ/4) e^(−i(Î{circumflex over (Z)}−{circumflex over (Z)}Î)δ) ^(−b) ^(/4) e^(−i(Î{circumflex over (Z)}+{circumflex over (Z)}Î)δ) ⁺ ^(/4) where Îrepresents the identity operator, {circumflex over (X)} represents thePauli-X operator, Ŷ represents the Pauli-Y operator, {circumflex over(Z)} represents the Pauli-Z operator, δ_(−a), δ_(−b) represent detuningoperation parameters, θ represents a swap angle for the swap operatorand ϕ represents a controlled-Z phase angle for the controlled-Zoperator.
 23. The method of claim 22, wherein determining the firstpulse, second pulse and third pulse using the control model comprises:determining values of detuning operation parameters, a swap angle and aphase angle such that the control model represents the action of thetwo-qubit quantum logic gate on the first qubit and the second qubit.24. The method of claim 23, wherein determining values of the detuningoperation parameters, swap angle and phase angle comprises setting thedetuning operation parameters, swap angle and phase angle to respectivevalues that have been determined through a calibration procedure, thecalibration procedure comprising: for each set of values of qubitcontrol parameters from multiple sets of values of qubit controlparameters: performing a cross-entropy benchmarking experiment using theset of values for the qubit control parameters to identify a unitarytransformation realized by the set of values of qubit controlparameters, comprising determining values of control model parametersthat maximize the fidelity of the identified unitary transformation;storing data linking the set of values of qubit control parameters, theunitary transformation the set of values realizes, and correspondingdetermined values of control model parameters; interpolating over thestored data to determine unitary transformations and the control modelparameters corresponding to sets of values of qubit control parametersthat are not included in the multiple sets of values of qubit controlparameters.
 25. The method of claim 24, wherein qubit control parameterscomprise qubit voltage, qubit coupler voltage and pulse length.
 26. Themethod of claim 17, wherein implementing the two-qubit quantum logicgate comprises implementing the two-qubit quantum logic gate withaccuracy above a predetermined threshold.
 27. An apparatus forimplementing a target two-qubit quantum logic gate, the apparatuscomprising: a first qubit coupled to a second qubit via a tunablecoupler; control electronics comprising one or more processors; and anon-transitory computer storage medium encoded with instructions that,when executed by the one or more processors, cause the controlelectronics to perform operations comprising: generate a control signalfor the target two-qubit quantum logic gate according to a controlmodel, wherein the control model comprises a controlled-Z operator and aswap operator that are non-orthogonal; and apply the control signal tothe first qubit, second qubit and tunable qubit coupler.
 28. Anapparatus for implementing a two-qubit quantum logic gate, the apparatuscomprising: a first qubit coupled to a second qubit via a tunable qubitcoupler; control electronics comprising one or more processors; and anon-transitory computer storage medium encoded with instructions that,when executed by the one or more processors, cause the controlelectronics to provide one or more control signals to the first qubit,second qubit, and tunable coupler to implement a controlled-Z operationand a swap operation, wherein the controlled-Z operation and the swapoperation are non-orthogonal.